Given relatively prime positive integers $a_1,\ldots,a_n$, the Frobenius number is the largest integer that cannot be written as a nonnegative integer combination of the $a_i$. We examine the parametric version of this problem: given $a_i=a_i(t)$ as functions of $t$, compute the Frobenius number as a function of $t$. A function $f:\mathbb{Z}_+\rightarrow\mathbb{Z}$ is a quasi-polynomial if there exists a period $m$ and polynomials $f_0,\ldots,f_{m-1}$ such that $f(t)=f_{t\bmod m}(t)$ for all $t$. We conjecture that, if the $a_i(t)$ are polynomials (or quasi-polynomials) in $t$, then the Frobenius number agrees with a quasi-polynomial, for sufficiently large $t$. We prove this in the case where the $a_i(t)$ are linear functions, and also prove it in the case where $n$ (the number of generators) is at most 3.
@article{10_37236_5112,
author = {Bjarke Hammersholt Roune and Kevin Woods},
title = {The parametric {Frobenius} problem},
journal = {The electronic journal of combinatorics},
year = {2015},
volume = {22},
number = {2},
doi = {10.37236/5112},
zbl = {1339.11042},
url = {http://geodesic.mathdoc.fr/articles/10.37236/5112/}
}
TY - JOUR
AU - Bjarke Hammersholt Roune
AU - Kevin Woods
TI - The parametric Frobenius problem
JO - The electronic journal of combinatorics
PY - 2015
VL - 22
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.37236/5112/
DO - 10.37236/5112
ID - 10_37236_5112
ER -
%0 Journal Article
%A Bjarke Hammersholt Roune
%A Kevin Woods
%T The parametric Frobenius problem
%J The electronic journal of combinatorics
%D 2015
%V 22
%N 2
%U http://geodesic.mathdoc.fr/articles/10.37236/5112/
%R 10.37236/5112
%F 10_37236_5112
Bjarke Hammersholt Roune; Kevin Woods. The parametric Frobenius problem. The electronic journal of combinatorics, Tome 22 (2015) no. 2. doi: 10.37236/5112
